| Abstract: |
| We construct some solutions for the fractional Yamabe problem which are singular along a prescribed set $\Sigma$. This is a problem which arises in conformal geometry when we try to find metrics that are conformal to the given one, have constant fractional curvature and are singular along the prescribed set. It is equivalent to finding positive and smooth solutions for
$$ (-\Delta)^\gamma u= c_{n, {\gamma}}u^{\frac{n+2\gamma}{n-2\gamma}}, u>0 \ \mbox{in}\ \r^n \backslash \Sigma.$$
The fractional curvature, a generalization of the classical scalar curvature, is defined from the conformal fractional Laplacian, which is a non-local operator constructed on the conformal infinity of a conformally compact Einstein manifold.
When the singular set $\Sigma$ is composed of one point, some new tools for fractional order ODEs can be applied to show that a generalization of the usual Delaunay solves the fractional Yamabe problem with an isolated singularity at $\Sigma$. (Joint work with Mar\`ia del Mar Gonz\`alez.)
If the set $\Sigma$ is a finite number of points, then, by using gluing methods, we will provide a solution for the fractional Yamabe problem with singularities at $\Sigma$. In order to preserve the non-locality of the problem, we need to glue infinitely many bubbles per point removed. This seems to be one of the first works in which a gluing method is successfully applied to a non-local problem. (Joint work with Weiwei Ao, Mar\`ia del Mar Gonz\`alez and Juncheng Wei.)
Finally, for a higher-dimension smooth submanifold $\Sigma$, we are able to adapt the classical gluing methods for the scalar curvature to the non-local setting. However, this requires us to develop new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. Some examples which are worth mentioning are the construction of radial fast-decaying solutions (using a blow-up argument and a bifurcation method); the use conformal geometry to rewrite this non-local ODE, giving a hint of what a non-local phase-plane analysis should be; and the study of a fractional Schr\{o}dinger equation with a Hardy type critical potential. (Joint work with Weiwei Ao, Hardy Chan, Marcos Fontelos, Mar\`ia del Mar Gonz\`alez and Juncheng Wei.) |
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